Multitapering for Estimating Time Domain Parameters of Autoregressive Processes

The most commonly used method for estimating the time domain parameters of an autoregressive process is to use the Yule-Walker equations. The Yule-Walker estimates of the parameters of an autoregressive process of order p, or AR(p), are known to often be highly biased. This can lead to inappropriate order selection and very poor forecasting. There is a Fourier transform relationship between the autocovariance sequence for an autoregressive process (the estimates of which are used in the Yule-Walker equations to estimate the time domain parameters) and the spectrum, a frequency domain representation of the autoregressive process. Tapering has been shown to reduce the bias of both the periodogram, a naive estimator of the spectrum, and of time domain parameter estimates. A new method, multitapering, has had great success in spectral estimation. We use the Fourier transform relationship between the frequency domain and the time domain to define multitaper Yule-Walker estimates of the time domain parameters of the autoregressive process, and evaluate their asymptotic distribution as well as their order 1/T bias, where T is the length of the series under consideration. We perform a Monte Carlo simulation of various autoregressive processes and compare the traditional Yule-Walker parameter estimates with single taper estimates and the new multitaper estimates. Our simulation results show that traditional single tapers perform as well as multiple tapers in time domain parameter estimation, despite the fact that multiple tapers perform better than single tapers in frequency domain estimation. We provide intuitive explanations as to why this might be so. keywords: autoregression; taper; Yule-Walker.